Irish Intervarsity Mathematics Competition 2002 University College Dublin Time allowed: Three hours
... 5. If f (n) = an2 + bn + c, where a, b, c and n are all positive integers, show that there exists a value of n for which f (n) is not a prime number. 6. What is the area of a smallest rectangle into which squares of areas 12 , 22 , 32 , 42 , 52 , 62 , 72 , 82 textand92 can simultaneously be fitted w ...
... 5. If f (n) = an2 + bn + c, where a, b, c and n are all positive integers, show that there exists a value of n for which f (n) is not a prime number. 6. What is the area of a smallest rectangle into which squares of areas 12 , 22 , 32 , 42 , 52 , 62 , 72 , 82 textand92 can simultaneously be fitted w ...
Notes 10
... In addition to the light shedding, that fact is of great importance in the theory and is repeatedly used. Theorem 2 Let G be a compact, connected Lie group of rank one. Then either G is isomorphisc to S1 or dim G = 3. Proof: Equip the Lie algebra Lie G with an inner product � v, w � invariant under ...
... In addition to the light shedding, that fact is of great importance in the theory and is repeatedly used. Theorem 2 Let G be a compact, connected Lie group of rank one. Then either G is isomorphisc to S1 or dim G = 3. Proof: Equip the Lie algebra Lie G with an inner product � v, w � invariant under ...
THE KEMPF–NESS THEOREM 1. Introduction In this talk, we will
... In this talk, we will prove the Kempf–Ness theorem which relates certain algebraic quotients with certain symplectic quotients. More precisely, let G be a complex reductive group acting linearly on a smooth complex projective variety X ⊂ PnC . Then one can consider the geometric invariant theory quo ...
... In this talk, we will prove the Kempf–Ness theorem which relates certain algebraic quotients with certain symplectic quotients. More precisely, let G be a complex reductive group acting linearly on a smooth complex projective variety X ⊂ PnC . Then one can consider the geometric invariant theory quo ...
Category of Compact Quantum Semigroups
... Suppose Γ is a group of integers Z. As an example of such semigroup S we could choose a semigroup of non-negative integers Z+ ⊂ Γ. The Pontryagin dual of Z, its group of characters, would be a unit circle G = S 1. For S = Z+, C astred(S) = T is the Toeplitz algebra. Note, that for the same group Γ w ...
... Suppose Γ is a group of integers Z. As an example of such semigroup S we could choose a semigroup of non-negative integers Z+ ⊂ Γ. The Pontryagin dual of Z, its group of characters, would be a unit circle G = S 1. For S = Z+, C astred(S) = T is the Toeplitz algebra. Note, that for the same group Γ w ...
First Class - shilepsky.net
... Similar situations occur when we study sets with binary operations. We talked about some structural properties. If we have the right combination of properties, important and useful algebraic structures result. What are some useful things we have done with algebra? Solving a simple equation in the in ...
... Similar situations occur when we study sets with binary operations. We talked about some structural properties. If we have the right combination of properties, important and useful algebraic structures result. What are some useful things we have done with algebra? Solving a simple equation in the in ...
Homomorphism of Semigroups Consider two semigroups (S, ∗) and
... (a) Let M be the set of all 2 × 2 matrices with integer entries. The determinant of any matrix ac bd A= is denoted and defined by det(A) = |A| = ad − bc. One proves in Linear Algebra that the determinant is a multiplicative function, that is, for any matrices A and B , det(AB) = det(A) · det(B) Thus ...
... (a) Let M be the set of all 2 × 2 matrices with integer entries. The determinant of any matrix ac bd A= is denoted and defined by det(A) = |A| = ad − bc. One proves in Linear Algebra that the determinant is a multiplicative function, that is, for any matrices A and B , det(AB) = det(A) · det(B) Thus ...
Representations of su(2) 1 Lie and linear groups
... representations in the space Vj of homogeneous polynomials in z, w ∈ C of degree 2j induced by the natural action of SU(2) on (z w)t ∈ C2 . This result can be established by restricting to the subgroup of diagonal matrices in SU(2) which is isomorphic to U(1), and showing that Vj splits as a sum of ...
... representations in the space Vj of homogeneous polynomials in z, w ∈ C of degree 2j induced by the natural action of SU(2) on (z w)t ∈ C2 . This result can be established by restricting to the subgroup of diagonal matrices in SU(2) which is isomorphic to U(1), and showing that Vj splits as a sum of ...
SHIMURA CURVES LECTURE 5: THE ADELIC PERSPECTIVE
... the case where the theory of quadratic forms tells us that any totally indefinite quaternary quadratic form over a number field F is universal, i.e., the map (character!) N : B × → F × is surjective. So certainly there exists some element of B of norm −1. A bit of classical number theory gives the f ...
... the case where the theory of quadratic forms tells us that any totally indefinite quaternary quadratic form over a number field F is universal, i.e., the map (character!) N : B × → F × is surjective. So certainly there exists some element of B of norm −1. A bit of classical number theory gives the f ...
Thompson`s Group F is not SCY
... manifold, by [Go95] and, pushing the dimension up by 2, of symplectic 6–manifolds with trivial canonical class by [FiPa11]. In spite of that, we will show that the constraints discussed above are sufficient to show that F is not SCY. The main difficulty lies in the fact that the constraint on the fi ...
... manifold, by [Go95] and, pushing the dimension up by 2, of symplectic 6–manifolds with trivial canonical class by [FiPa11]. In spite of that, we will show that the constraints discussed above are sufficient to show that F is not SCY. The main difficulty lies in the fact that the constraint on the fi ...
Mathematics Qualifying Exam University of British Columbia September 2, 2010
... a) Give the first three nonzero terms for the Laurent expansion of f (z) about 12 . b) Give the first three nonzero terms for the Laurent expansion of f (z) about 0, valid for small |z|. Give the region of convergence for the full expansion. c) Give the first three nonzero terms for the Laurent expa ...
... a) Give the first three nonzero terms for the Laurent expansion of f (z) about 12 . b) Give the first three nonzero terms for the Laurent expansion of f (z) about 0, valid for small |z|. Give the region of convergence for the full expansion. c) Give the first three nonzero terms for the Laurent expa ...
Physics 880K20 (Quantum Computing): Problem Set 1. David Stroud, instructor
... Show that the requirement that (A’, B’) = (A, B) requires that the square matrix U is unitary, i. e., that U† U = I, where I is the identity matrix. (n × n if A and B have n components). 2. Consider a system described by a time-independent Hamiltonian. The solution of the time-dependent Schrodinger ...
... Show that the requirement that (A’, B’) = (A, B) requires that the square matrix U is unitary, i. e., that U† U = I, where I is the identity matrix. (n × n if A and B have n components). 2. Consider a system described by a time-independent Hamiltonian. The solution of the time-dependent Schrodinger ...
Towards a Deformation Quantization of Gravity
... ⋆ (ℏ) in the space of associative bilinear products. ...
... ⋆ (ℏ) in the space of associative bilinear products. ...
Symplectic Topology
... and classify groups acting locally on R k for which (i) the group acts locally transitively (or we could just reduce dimension to an orbit) (ii) the group has no invariant “foliation”: it’s not of the form (x, y) 7→ φ(x, y) = (f (x), g(x, y)) for R k = R l × R k−l (or simplify by φ 7→ f ). Theorem ( ...
... and classify groups acting locally on R k for which (i) the group acts locally transitively (or we could just reduce dimension to an orbit) (ii) the group has no invariant “foliation”: it’s not of the form (x, y) 7→ φ(x, y) = (f (x), g(x, y)) for R k = R l × R k−l (or simplify by φ 7→ f ). Theorem ( ...
